In part 1 (see http://sigma5.blogspot.com/2017/08/principia-part-1.html) I talked around Principia, Newton's foundational book about Celestial Mechanics and many other things. In this post I am going to talk about what's in it. But first a digression. I am going to briefly discuss Analytic Geometry and Calculus. DON"T PANIC! The discussion will be almost entirely math free. I just want to introduce some ideas that give you a basic feel about that each subject is about.
Starting with Analytic Geometry, imaging a piece of graph paper. This is the common kind where it is just a piece of paper full of square boxes. Now let's number each column along the top and each row down the side. Now pick any box. If we sight down the column of boxes containing our special box we can see the column number our box is in. If we sight across the page along the row boxes we can see the row number our box is in. If we list the row number and column number it uniquely identifies our special box.
Now imagine a flat piece of paper. This time the paper starts out blank so there are no boxes and no row numbers and no column numbers. But we can do a more sophisticated version of the same thing. We draw a horizontal straight line across the page. Then we draw a straight line that is perpendicular to the original line down the page. It will cross our original line at a point called the Origin. Then we use a ruler to mark a distance scale off on each of these lines. We can now uniquely identify any single point on our piece of paper by citing it's "coordinates". By convention the horizontal line is the "X axis" and positive distances go off to the right. And by convention the vertical line is the "Y axis" and positive distances go up toward the top. And assume the piece of paper is as big as we need it to be.
Now pick a point, any point, as long as it is not on either of our axes. Now draw a straight line that is parallel to the X axis through our special point. It will strike the Y axis at some point. Draw a second straight line through our special point parallel to the Y axis. It will strike the X axis at some point. Read the distance along the X axis to the point of intersection and turn it into a number by using our scale. Do the same thing with the Y axis. This yields two numbers, conventionally recorded as say (3.74,-8.23), the coordinates of the point. These coordinates uniquely identify the location of the point. And this process can be used to determine the coordinates of any point.
Now we can turn things into algebra. Instead of talking about the point (3.74, -8.23) we can talk about a point "at (x,y)". And we can write algebraic equations involving x and y. By convention, w, x, y, and z represent "variables", numbers whose values we may or may not know, and a, b, c d, represent "constants", numbers we at least in theory know the value of. And we may allow the value of a variable to vary. Okay, on we go.
Consider a very simple equation: x = 0. In Analytic Geometry we play around with equations like this. They are always of the form (something) = (something else). We ask the question "what are all the possible values of "y" that are consistent with our equation "x = 0" being true?". Technically we ask "what is the locus of solutions for the equation 'x = 0'?" A locus is just a bunch of points for which the equation in question is true. And we can graph this particular locus by marking all the points in the locus on the paper. It turns out to just be the Y axis. Similarly the X axis is the locus of solutions to the equation "y = 0". And that's the fundamental idea behind Analytic Geometry.
We can turn diagrams, the stuff Geometry has been about, into equations. The reason we want to do this is there are a whole bunch of tricks from Algebra that we can now apply to solving problems. Now consider the equation "x squared + y squared = 1". It turns out if we graph the locus of solutions to this equation we get a circle around the Origin with a radius of 1. In fact the graph of "x squared + y squared = r squared" gives us a circle with a radius of "r" (assumed to be a constant in this discussion).
And it turns out that there is an algebraic formulation for an ellipse, a parabola, etc. And I have given the equation for a circle centered at the Origin. There is a more complicated equation for a circle whose center is located elsewhere. This is also true for an ellipse, etc. But that's making things more complicated than I want to get into so I'm going to skip all that. Because that's all I am going to say about Analytic Geometry. See, that wasn't so hard. On to Calculus.
Consider the equation "y = x". If we graph its locus we get a diagonal line through the Origin going up and to the right (and also down and to the left) at a 45 degree angle. Now let's say we want to calculate the area "under the curve" (and a straight line is a kind of curve) for "y = x" where x goes from 0 to 100. It turns out that there is a simple way to do this that the Greeks figured out about 3,000 years ago. But we are going to ignore that. (We are going to do a lot of ignoring for the rest of this discussion.) Let's divide things up into columns, one column for each inch (I am going to assume our scale is in inches to make the explanation easier to follow). Consider the first column, the one going from x=0 to x=1. It turns out we end up with a cute little triangle that is 0 inches high on the left (x = 0) and 1 inch high on the right (x = 1). Let's set this triangle aside for the moment.
Now consider the next column. It is 1 inch high on the left side and 2 inches high on the right. This can be subdivide into a nice 1 inch by 1 inch square and another of those pesky triangles. Let's set the triangle part aside. We can now put the square into a bucket called "part of the solution". Now we move on to the next column. Using the same process we end up with a third pesky triangle and a rectangle that is two inches high and one inch wide. It's area is obviously two square inches. So let's add that two square inches into our "part of the solution" bucket and move on.
We keep doing this. We end up with 100 pesky triangles and a bunch of rectangles. What do all the rectangles add up to? Let's just assume we have a method for figuring this out and ignore what both the method is and what the answer is. Now consider our pesky little triangles. The total area under the curve is what those rectangles add up to plus what those triangles add up to. Assuming the triangles add up to something, which they obviously do, then if we preliminarily take the answer as just the sum of the rectangles then we know this answer is wrong. It is low by just the amount that the triangles add up to. All this seems needlessly complicated. And for the toy problem we are considering it is. But all will soon become clear.
What we have is a tentative answer (the sum of the rectangles) plus some amount of error (whatever the triangles add up to). How much error? Well, let's turn those pesky triangles into squares that completely contain the triangles. We can add these squares together easily because there are 100 of them and each of them is 1 inch by 1 inch. So we know that our tentative answer is within 100 square inches of the correct one. It looks like at this point we really haven't gotten anywhere but appearances are deceiving.
Let's go through the same process again. But instead of using 1 inch columns let's use half inch columns. We are going to end up with a bunch of rectangles that are a half an inch wide and some number of inches high. That should make it harder to figure out how much they add up to. But let's assume there is some procedure for figuring this out and ignore it. Instead let's focus on the error. And let's again do the same thing where we turn the triangles into squares. Now we have 200 of them. That sounds bad. But each of them is half an inch on a side so each of these new squares has an area of only a quarter of a square inch. Our maximum error has gone from 100 square inches to 50 square inches (200 squares times a quarter of a square inch per).
Now let's cut the width in half again so it is a quarter inch. The result is that our maximum error is again cut in half to 25 square inches. Okay we are now where we need to be. Let's keep halving the width over and over and over. Every time we do we cut our maximum error in half. We can keep doing this almost forever. If we do it forever we end up with a width of zero and we run into "divide by zero" problems. So instead let's keep doing it almost forever. The width keeps getting smaller and smaller but we assume it never quite makes it to zero. This "as small as we want it but never exactly zero" is called an "infinitesimal". And Newton used the word "fluxion", a word that obviously did not catch on, for what we now call an infinitesimal.
And this business of slicing things up more and more until you get to an infinitesimal is called "taking the limit". If we can drive the maximum error below any arbitrary number no matter how small then we can use our technique for adding together all those rectangles to get an answer that is for all intents and purposes exactly the correct answer. And that's Integral Calculus.
Differential Calculus works along the same lines. You find an "approximate" method for calculating the tangent along each point on the line. We again have an unspecified way (in the sense that I am not going to specify it) of coming as close as we want to the actual value of the tangent in a manner similar to adding up all the rectangular columns. Then you slice things finer and finer until you are down to infinitesimals (but not zero). If there is such a method then that's the method for "differentiating" whatever equation you stared with.
There is lots of fine print, ways for this to go wrong, but mathematicians have figured out the characteristics necessary to guarantee that an equation can be integrated or differentiated. And now I'll let you in on the other big Calculus secret. It's all a bag of tricks. If an equation has this certain form then this trick works. If it has this other certain form then this other trick works. There are lots of tricks for handling equations that have lots of forms. And this allows Calculus to solve for areas or slopes of all kinds of curves. And both Calculus and Analytic Geometry are easily extended to handle more than the two dimensions (i.e. "X" and "Y" and "Z" and maybe "W" and . . .). But I am going to save you from all that by not talking about it.
This business of limits and infinitesimals preceded Newton. But what Newton brought to the table was a bunch of new tricks that allowed this limits/infinitesimals business to be used to solve much more difficult problems than his predecessors had cracked. He needed his Calculus, his bag of tricks, to be able to perform the calculations necessary to reach the solutions he needed. He was a very smart man. And scattered all through Principia are tricks for handling this, that, or the other kind of equation. So that's one of the things that is in Principia. But this whole Calculus business was a means to an end for Newton. So what were some of the ends?
Well, here's another digression. But this time it's not my digression. It's Newton's. The first two sections of Principia (it is divided into three main sections) were strictly mathematical. He says in effect "let's ignore the real world for a while and for the moment just assume that a certain mathematical formulation is how the world works and use that to figure out what things would look like".
So he assumed that things worked like he thought gravity worked. He then went on to show that planetary orbits would be ellipses (or in some cases circles or parabolas). But then he said that things would work this other way if a different mathematical formulation was assumed to be how the world worked. That gave him the mathematical foundation to explore alternatives to his description of how Celestial Mechanics worked. And he was very thorough. He explored mathematically a number of different models. He concluded in section 3 that his formulation matched how gravity worked in the real world and that the alternatives did not.
And there is a reason it is called "Newton's theory of universal gravitation". In Newton's time conventional wisdom held that there was one set of rules for earth and things near the ground (birds, mountain tops) and a different set of rules for the heavens (the sun, moon, planets, stars, etc.) Newton said "nope". Gravity works the same everywhere. He calculated what the force of gravity should be between the earth and the moon. He showed that it was the same (except that it diminished with distance exactly as he described) gravitational force was just right to keep the moon circling the earth. He did the same thing for the sun and the planets. Then he did the same thing for the moons of Jupiter and Saturn and for comets. That completely demolished the idea that there were different laws for near earth and up in the heavens.
But he went further, much further. He calculated how much people would weigh if they traveled from the surface of the earth to the center of the earth. Scientists has speculated that the earth was not a perfect sphere. Newton calculated the amount the earth would be distorted as a result of the fact that it rotated once per day. The technology of the day was not accurate enough to confirm this. But the technology of the day was good enough to show that gravity was slightly less at the top of a mountain than it was at sea level. Newton calculated exactly how much.
In Celestial Mechanics there is something called the "two body" problem. What is the path of two bodies orbiting each other if only the force of gravity between the two is important. Newton solved that. And he tackled the "three body" problem. Specifically he looked at a system consisting of the earth, moon, and sun. He was able to calculate the "perturbations" caused by the effect of the sun on the orbits around each other of the earth and the moon.
As a corollary to this investigation he was able to explain the tides. They are semi-periodic. He showed that the moon's gravitational pull on different parts of the earth caused a "lunar" tide (the larger) and the sun's gravitational effect on different parts of the earth caused a "solar" tide (smaller but significant). They would move in phase and out of phase resulting in the complex pattern of high tides and low tides we see. This was the first successful theory of tides.
He also tackled the mutual perturbations of Jupiter on Saturn's orbit and Saturn on Jupiter's orbit. Before Newton no one had even tried to do that.
He investigated the motions of pendulums. It turns out that pendulums are a great way to accurately measure the force of gravity. They had been used, for instance, to investigate the force of gravity on the tops of mountains (see above). He also put forward mathematical formulations of drag, how bodies moved through fluids like air and water. These were pioneering studies. They were not completely successful (we now know that the situation is quite complicated). But he was able to show by his theoretical work and also by some experimental work that a number of then current theories were wrong.
And, of course, there was the business of winning arguments. As I mentioned in the previous post, Descartes had put forward a "vortex" theory of gravity. Newton's work on the orbits of Jupiter, Saturn, and especially comets, knocked the theory to flinders. But Newton was nothing if not thorough. So he did a mathematical treatment of vortexes (that was one of the mathematical alternatives he explored) that showed that vortexes did not work for mathematical reasons. But that was still not enough. Destroying the vortex theory was one of the reasons he explored drag. For vortexes to work there had to be some kind of fluid. That is how the vortex associated with one body influenced the path of another body. But his mathematical study of drag indicated that there was no way to get vortexes to create the right amount of drag to exert the right amount of force in the right place to make each body move in the way it was observed to move.
Then there was the "epicycle" theory, the traditional method of dealing with the orbits of planets. Originally the idea was that the planets moved in perfect circles. But that didn't work. So the idea was you had a primary circle. Then there was a secondary circle. The planet was attached to the secondary circle which was in turn attached to the primary circle. That didn't quite work either so eventually there were models involving large numbers of "epicycles" (these secondary circles) all connected together in a complex way. Newton developed mathematically a celestial mechanics of epicycles. This was yet another of the alternative mathematical systems he explored. Then he showed that the planets just didn't move according to his epicycle mechanics system, no matter how you arranged the epicycles.
Finally let me finish up with some more general observations. Principia is very hard to read. That's why I skimmed most of it just deeply enough to see what was being discussed. Besides the reasons I have already discussed there is the "terminology" problem. A lot of what Newton was dealing with was completely new. So he invented terminology that got replaced with other terminology in the intervening centuries. For instance what he called fluxions we now call infinitesimals. That's pretty straight forward as our modern term means the same thing as his term.
But Newton introduced the idea of mass. Mass is an inherent property of matter. It is the degree with which it resists being accelerated by a given force. Newton never did just introduce a word like "fluxion" for this concept. So he ends up talking around what he is at in a way that is confusing. To a greater or lesser extent the same is true with "inertia", impulse", "momentum", energy", "work", and others.
These are terms that now have well established definitions and usages. But in some cases Newton was dealing with the concept for the first time because he had invented it. In other cases had an unclear or incomplete idea of what eventually ended up to be the modern concept. This made him understandably sloppy with usage. In some cases he didn't even know that the concept might exist. This makes it hard to follow a lot of his arguments. You have to go through a process of translating what Newton says into what he means. Then you must make allowances for the fact that he did not have a clear idea of how the concept worked. If Newton had understood these concepts the way we now do he could have laid out his logic much more quickly, more easily, and in a more understandable way.
I ended my previous post with a note on Newton and theology. If you take the whole of his life into account theology was more important and played a bigger part than his scientific work. The same is true of his great rival Descartes. The scientific work we now remember Descartes for was also done in his youth. But he too spent a much greater part of his life on theological issues. I can't speak to Descartes's theological ideas. But I can speak to Newton's. He spent a little time discussing them at the very end of Principia. But he also went into them at some length in Optics.
Newton believed there was two kinds of truth. There was what we now call scientific truth, what was then called natural philosophy. Then there was religious truth. Newton's belief was that these two kinds of truth were not antagonistic but complementary. Combined appropriately, they resulted in a kind of super-truth that was more powerful than either of them standing on its own.
That was the main theological problem he spent his time on. He could see that "old time religion" theology just did not work. So he tried to tweak mainstream theology to produce something that retained the important parts of mainstream religious thinking but resulted in something that was compatible with science. I don't know to what extent he thought he succeeded. What I do know is that Newton's scientific work survives and is hugely influential. But his religious ideas have vanished without a trace. And one reason I know nothing about Descartes's religious thinking even though I am familiar with his scientific work is because his work on theology vanished without a trace soon after his death too.
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